Initial condition for wave equations

[kamensky]

You can use

u1.assign(ic)

if ic is a Function in V (as the output of interpolate would be). The argument may also be a constant-coefficient linear combination of Functions in V. Setting u0 depends on having another initial condition for the velocity, or knowing u(t_0-\Delta t) directly.

You might also consider re-arranging the time integrator to use a displacement and velocity from the previous time step, as implemented in my answer here:

Second order time discretization problem

You can convert to a first order system but then formally eliminate the extra velocity unknown to avoid solving in a mixed function space. The implicit midpoint rule gives you \frac{u^{n+1} - u^n}{\Delta t} = \frac{1}{2}(\dot{u}^{n+1} + \dot{u}^{n})~~~\Rightarrow~~~\dot{u}^{n+1} = 2(\frac{u^{n+1} - u^n}{\Delta t}) - \dot{u}^n which can be substituted into the left-hand side of \frac{\dot{u}^{n+1} - \dot{u}^n}{\Delta t} = F((u^{n+1} + u^{n})/2) to get a problem for u^{n+1} alone, given bot…

Another note: If you initialize the wave equation with discontinuous data, you will likely get spurious oscillations in the solution without some form of additional stabilization or shock capturing. (That was one of the demonstrative examples considered in this paper. FEniCS code for the wave equation test is here, but it uses an additional library I wrote to do isogeometric analysis with FEniCS.)